Neglected X-ray discovered polars: II. The peculiar eclipsing binary HY Eridani

We report on the X-ray observations of the eclipsing polar HY Eri (RX J0501-0359), along with its photometric, spectrophotometric, and spectropolarimetric optical variations, collected over 30 years. With an orbital period of 2.855 h, HY Eri falls near the upper edge of the 2-3 h period gap. After 2011, the system went into a prolonged low state, continuing to accrete at a low level. We present an accurate alias-free long-term orbital ephemeris and report a highly significant period change by 10 ms that took place over the time interval from 2011 to 2018. We acquired a high-quality eclipse spectrum that shows the secondary star as a dM5-6 dwarf at a distance d=1050 +/- 110 pc. Based on phase-resolved cyclotron and Zeeman spectroscopy, we identify the white dwarf (WD) in HY Eri as a two-pole accretor with nearly opposite accretion spots of 28 and 30 MG. The Zeeman analysis of the low state spectrum reveals a complex magnetic field structure, which we fit by a multipole model. We detected narrow emission lines from the irradiated face of the secondary star, of which Mg I lambda 5170 with a radial velocity amplitude of K-2 = 139 +/- 10 km s(-1) (90% confidence) tracks the secondary more reliably than the narrow H alpha line. Based on the combined dynamical analysis and spectroscopic measurement of the angular radius of the WD, we obtain a primary mass of M-1=0.42 +/- 0.05 M-circle dot (90% confidence errors), identifying it as a probable He WD or hybrid HeCO WD. The secondary is a main sequence star of M-2=0.24 +/- 0.04 M-circle dot that seems to be slightly inflated. The large distance of HY Eri and the lack of similar systems suggest a very low space density of polars with low-mass primary. According to current theory, these systems are destroyed by induced runaway mass transfer, suggesting that HY Eri may be doomed to destruction. Over the last 30 years, HY Eri experienced high and low states with mass transfer rates that differed by three orders of magnitude, varying between M similar or equal to 10(-9)M(circle dot) yr(-1) and 10(-12)M(circle dot) yr(-1). At a galactic latitude of -26.1 degrees, it is located about 500 pc below the galactic plane.